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The problem is to express the quadratic polynomial \(1 + 4x - 2x^2\) in the form \(p - q(x + r)^2\) where \(p\), \(q\), and \(r\) are real numbers.

To find the values of \(p\), \(q\), and \(r\), we rewrite \(1 + 4x - 2x^2\) in the desired format. We start by expanding the form \(p - q(x + r)^2\):

\[
p - q(x + r)^2 = p - q(x^2 + 2rx + r^2) = p - qx^2 - 2qrx - qr^2
\]

Now we set this equal to \(1 + 4x - 2x^2\) and equate the coefficients for each power of \(x\):

\[
- qx^2 - 2qrx - qr^2 + p = 1 + 4x - 2x^2
\]

From this, we set up the following system of equations:

1. Coefficient of \(x^2\): \( -q = -2 \)
2. Coefficient of \(x\): \( -2qr = 4 \)
3. Constant term: \( -qr^2 + p = 1 \)

From the first equation, we immediately find:
\[
q = 2
\]

Substitute \(q = 2\) into the second equation to find \(r\):
\[
-2 \times 2 \times r = 4 \implies -4r = 4 \implies r = -1
\]

Using \(q = 2\) and \(r = -1\) in the third equation to solve for \(p\):
\[
-2 \times (-1)^2 + p = 1 \implies -2 + p = 1 \implies p = 3
\]

Thus, the expression in the desired form is:
\[
3 - 2(x - 1)^2
\]

Would you like to know more about this method or do you have any other questions? Here are some related topics:

1. How does completing the square work for rewriting quadratic expressions?
2. What happens if you change the sign of \( q \) in the expression?
3. How can we verify the correctness of the expression \(3 - 2(x - 1)^2\)?
4. Can this method be applied to higher degree polynomials?
5. What are the graphical interpretations of the form \(p - q(x + r)^2\)?

Tip: Always check if the discriminant can help simplify solving quadratic equations, especially when dealing with expressions that need to be rewritten.

Express 1 + 4x - 2x^2 in the form p - q(x + r)^2, where p, q, and r are real numbers.

Learn how to express the quadratic equation 1 + 4x - 2x^2 in the form p - q(x + r)^2. This solution involves completing the square and equating coefficients to find p, q, and r. Suitable for high school students (Grades 10-12).

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